Continuity of asymptotic entropy on wreath products

We prove the continuity of asymptotic entropy as a function of the step distribution for non-degenerate probability measures with finite entropy on wreath products $A \wr B = \bigoplus_B A \rtimes B$, where $A$ is any countable group and $B$ is a countable hyper-FC-central group that contains a finitely generated subgroup of at least cubic growth. As one step in proving the above, we show that on any countable group $G$ the probability that the $\mu$-random walk on $G$ never returns to the identity is continuous in $\mu$, for measures $\mu$ such that the semigroup generated by the support of $\mu$ contains a finitely generated subgroup of at least cubic growth. Finally, we show that among random walks on a group $G$ that admit a separable completely metrizable space $X$ as a model for their Poisson boundary, the weak continuity of the associated harmonic measures on $X$ implies the continuity of the asymptotic entropy. This result recovers the continuity of asymptotic entropy on known cases, such as Gromov hyperbolic groups and acylindrically hyperbolic groups, and extends it to new classes of groups, including linear groups and groups acting on $\mathrm{CAT}(0)$ spaces.